Finite Square-Well Potential

24. Apply the boundary conditions to the finite square-well potential at x=0 to find the relationships between the coefficients A, C, and D and the ratio C/D.

I understand the wave equations in the three separate regions. For this question I need to only consider I, II. The wave equations need to decrease to zero as x approaches positive or negative infinity. The wave equation and its derivative need to be continuous as well. Thus, the wave equation of I equals II.

My professor did a similar problem last semester, but I can't make sense of his procedure. I think the delta function is in it, etc.

http://i111.photobucket.com/albums/n...g?t=1283054054

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Your professor wrote the two equations that correspond to
 Quote by Shackleford The wave equation and its derivative need to be continuous as well.

 Quote by vela Your professor wrote the two equations that correspond to
He did it a bit differently, though.

I also re-worked my part, too.

http://i111.photobucket.com/albums/n...g?t=1283054054

He has

1 = A + B
K = ik (A - B

ik/K = (A + B)/(A - B)

(ik + K)/(ik - K) = A/B = delta

I understand the manipulation up until here. I still don't know how in the heck this helps me related A, C, and D. Do I do the same thing?

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Finite Square-Well Potential

I think you're getting your coefficients mixed up. Your equations should be

A = C+D
αA = ik(C-D)

I'm not sure what your professor is doing. It looks like his A and B are your C and D and his K is your alpha. He took (your) A to be equal to 1 for some reason. The delta is not the delta function. It's just the quantity which equals A/B.

 Quote by vela I think you're getting your coefficients mixed up. Your equations should be A = C+D αA = ik(C-D) I'm not sure what your professor is doing. It looks like his A and B are your C and D and his K is your alpha. He took (your) A to be equal to 1 for some reason. The delta is not the delta function. It's just the quantity which equals A/B.
Oops. I accidentally wrote down B + C for some reason.

Here's what the professor did last week. I assume the book is looking for something like this. I don't know how he got this.

http://i111.photobucket.com/albums/n...g?t=1283058414

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus You have two equations and three unknowns, so you can solve for two of them, say C and D, in terms of the other, A. That's what your professor did except in his case, there were four unknowns, so he solved for B and C in terms of A and D.

 Quote by vela You have two equations and three unknowns, so you can solve for two of them, say C and D, in terms of the other, A. That's what your professor did except in his case, there were four unknowns, so he solved for B and C in terms of A and D.
Okay. I'll play around with the equations. Maybe I'll get partial credit. lol.

Oh, for the 1 = A + B, I think he used the free particle solution for the wave heading from the negative x direction towards the potential.

 Quote by vela You have two equations and three unknowns, so you can solve for two of them, say C and D, in terms of the other, A. That's what your professor did except in his case, there were four unknowns, so he solved for B and C in terms of A and D.
Well, I tried to play around with what the professor did, but I couldn't get his equations. Forgive my lack of algebraic-manipulation skills.

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Rewriting the equations a bit, you get \begin{align*} B - C & = -A + D \\ k_0 B + kC &= k_0 A + kD \end{align*} Multiply the first equation by k, add it the second, and solve for B.

 Quote by vela Rewriting the equations a bit, you get \begin{align*} B - C & = -A + D \\ k_0 B + kC &= k_0 A + kD \end{align*} Multiply the first equation by k, add it the second, and solve for B.
Okay. That makes sense. I'm still not sure about my problem. Playing with it, I got

A = [(ik + α)C + (α - ik)D] / 2α

Is that what the book is looking for?

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Probably not. Try solving for C and D in terms of A. Then you can calculate the ratio C/D.

 Quote by vela Probably not. Try solving for C and D in terms of A. Then you can calculate the ratio C/D.
Crap. I forgot about the ratio C/D. Well, maybe I'll get partial credit. The homework was due today.

Just now, I got A = [-2ik/(α-ik)] D.

I suspect C would be something similar.

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus I think that's right, and C comes out similarly.

 Quote by vela I think that's right, and C comes out similarly.
Well, that's just great. I do something right after it's due. I'm not going to do well on this problem set. Pretty much the rest of the problem sets will come from this book. I don't expect to do well on any of them.

http://www.amazon.com/Quantum-Physic.../dp/0471057002

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus If you don't like that text, you might want to see if your library has Griffith's book on quantum mechanics. I'll admit I've never seen it, but I used his particle physics book as an undergrad. He was very good at explaining concepts and showing how to apply them in problems. http://www.amazon.com/Introduction-Q...ref=pd_sim_b_1 (It's the top-selling book on quantum mechanics at Amazon.)

 Quote by vela If you don't like that text, you might want to see if your library has Griffith's book on quantum mechanics. I'll admit I've never seen it, but I used his particle physics book as an undergrad. He was very good at explaining concepts and showing how to apply them in problems. http://www.amazon.com/Introduction-Q...ref=pd_sim_b_1 (It's the top-selling book on quantum mechanics at Amazon.)
You think it'll help me do the Gasiorowicz problems? Even the professor said Gasiorowicz isn't an ideal textbook. Also, I saw one of my friend's particle physics book, and it's Griffiths, too.

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Yes, I think it would help. It couldn't hurt. Textbooks are so expensive, though, so I'd try to look through a copy in a bookstore or at the library first.